The Harris County Sheriff's Office said the incident occurred in and around the LSC-CyFair Health Science Center about 11:20 a.m. local time, and the campus was locked down and evacuated.
Sheriff's spokesman Thomas Gilliland said Quick, 21, a student at the Cy-Fair campus, was subdued by another student, KPRC-TV, Houston, reported.
Quick was born deaf and received a cochlear implant at age 7, the TV station said.
Six victims were taken to Memorial Hermann Texas Trauma Institute, the Houston Chronicle said. Two patients were in critical condition and three others had been upgraded to good, while one was discharged.
Nine other victims were taken to North Cypress Medical Center with minor injuries, Fire Department Capt. Robert Rasa said.
Gilliland said the stabbings appeared to be random.
"He came running and swinging at my neck, as I tried to get out of the way," said Michelle Alvarez, who suffered a slash to her neck.
Demond Lago, who was arrested by campus police for trespassing shortly before the incident, told the Chronicle he was sitting next to the suspect in the holding area.
Lago said he asked the suspect, a white male with red hair and freckles, what he was trying to do.
"He said he was trying to go on a killing spree but the [expletive] blade broke," Lago said.
Police said Lone Star sophomore Ryan Ballard tackled the suspect with the help of another student and restrained him until police arrived.
"I'm a pretty big dude," the 6-foot-4, 235-pound Ballard told the Chronicle. "I pushed him down with my elbow and I had my knee on his back while he [the other student] was holding his [the suspect's] face in the ground."